Chain Rule
Chain Rule
Objectives
- State the chain rule for the composition of two functions.
- Apply the chain rule together with the power rule.
- Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
- Recognize the chain rule for a composition of three or more functions.
- Describe the proof of the chain rule.
Summary
A composite function is one where the input variable \(x\) first passes through one function, and then the resulting output passes through another. For example, the function \(h(x) = 2^{\sin(x)}\) is composite since \(x \longrightarrow \sin(x) \longrightarrow 2^{\sin(x)}\text{.}\)
Given a composite function \(C(x) = f(g(x))\) where \(f\) and \(g\) are differentiable functions, the chain rule tells us that
\begin{equation*} C'(x) = f'(g(x)) g'(x)\text{.} \end{equation*}